
The convergence of sums in Eq. (REF ) follows from the fact that
\( \mu \)  is translation-invariant, \( d=1\) , and the variance of \( \mathfrak {s}([- R, R])\)  under \( \mu \)  increases logarithmically as \( R\rightarrow \infty \) .
The translation invariance of \( \mu \)  is clear because the determinantal kernel \( \mathcal {K} _{\mathrm {sin}, 2}\)  in Eq. (REF ) defining \( \mu \)  is translation-invariant. The logarithmic growth of the variance follows from a direct calculation using the Fourier transform [1]}.
The second equality in Eq. (REF ) comes from \( d = 1 \)  and the translation invariance of \( \mu \) .
